- Start with the equations \(y = 4x + 1\) and \(2y = -x - 1\).
- Substitute \(4x + 1\) for \(y\) in the second equation, resulting in \(2(4x + 1) = -x - 1\).
- Simplify the equation by distributing the 2 to get \(8x + 2 = -x - 1\).
- Combine like terms to reach the final form \(\boxed{9x = -3}\), which corresponds to option (c).
Explanation
- Problem Analysis and Objective
We are given two equations:
(1) \(y = 4x + 1\)
(2) \(2y = -x - 1\)
Our objective is to use the substitution method to replace \(y\) in the second equation with the expression for \(y\) from the first equation, then simplify to find the resulting equation. This is a standard algebraic technique to reduce a system of equations into a single-variable equation.
- Performing the Substitution
To substitute the expression \(y = 4x + 1\) into the equation \(2y = -x - 1\), we replace the \(y\) in the second equation with the quantity \((4x + 1)\).
This gives us:
\[2(4x + 1) = -x - 1\]
Now, apply the distributive property to the left side:
\[2 \times 4x + 2 \times 1 = -x - 1\]
\[8x + 2 = -x - 1\]
- Simplification and Final Equation
To simplify the equation \(8x + 2 = -x - 1\), we want to group all terms involving \(x\) on one side and the constant terms on the other.
Add \(x\) to both sides of the equation:
\[8x + x + 2 = -1\]
\[9x + 2 = -1\]
Next, subtract 2 from both sides to isolate the \(x\)-term:
\[9x = -1 - 2\]
\[9x = -3\]
This matches option (c).
Examples
The substitution method is essential in various real-life scenarios, such as budget planning or optimizing resource distribution. For instance, if you are planning an event where the total cost is constrained by a fixed budget and the price of one item depends on the number of guests, you can express the total cost as a function of guests and then substitute that into your budget equation to find exactly how many guests you can afford. Understanding this skill helps in solving everyday problems where variables are linked by specific mathematical rules.